Mathematical Sciences: Investigations in Spectral & Fractal Geometry: Vibrations of Fractal Drums, Spectral Zeta Functions, Analysis on Fractals, & Variational Ellip

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Lapidus will investigate sharpened forms of the asymptotic formula for the eigenvalue distribution of Laplacians with various boundary conditions. The goal is to make yet clearer the role of the Minkowski dimension of the boundary in this distribution. There are analogous formulas for the trace of the associated heat semigroup. The case in which not only the boundary of the domain, but the domain itself, is fractal will be studied. The relationship between the geometry of a manifold and the invariants of the elliptic differential operators on it is one of the main unifying themes of contemporary mathematics. The case of a region with fractal boundary, so irregular that its dimension is no longer an integer, has been the setting for some recent breakthroughs relating spectra information to geometry. This work has applications to the study of porous media and the scattering of waves from fractal surfaces.